If the unknown x is in the index, you can try applying logarithm to both sides of the equation.
2^x = x^3
log 2^x = log x^3
x log 2 = 3 log x
which would need to be solved by numerical methods to approximate the solution.
You can try to plot the graphs of the 2 equations:
{ y = 2^x
{ y = x^3
and you will see there is one solution between 1 and 2